Sounds like the derailleur to me as well and the easiest to try first.
But... Your chain could be all beat up and about to break, or your 'early riding' may have even damaged some teeth on one or two of your rear wheel sprockets, most likely gear 2 and/or 3 as typically those are the first 2 gears you use to start with and beat the hardest.
A powerful 2-stroke motor like say a 66cc with an expansion chamber plays **** on the whole drive train including the Jackshaft parts, but they can really rip a derailleur to pieces in short order, especially a 'budget bike' one.
I just popped in here to post some info and try to get a little info for my new project to takes some of that 2-stroke pulse-pounding out of the whole drive train on a Jackshafted bike, but I always try to help with a few answers or help from other posts before I ask my own.
In short, if I can get the design of this right, this is a replacement 'torque dampener' sprocket for the first (left side) sprocket on a Jackshaft kit.
The main object is for that orange rubber like material mellows out the motors power pulses before it even hits the drive train.
More about this when I post my topic here about it.